Question

use the unit circle to find the value of sin(7π/4) and periodic properties of trigonometric functions to find the value of sin(23π/4). select the correct choice below and fill in any answer boxes in your choice. a. sin(7π/4)=□ (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the solution is undefined. select the correct choice below and fill in any answer boxes in your choice. a. sin(23π/4)=□ (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the solution is undefined.

Explanation:

Step1: Rewrite $\frac{7\pi}{4}$

$\frac{7\pi}{4}=2\pi-\frac{\pi}{4}$. The sine - function has a period of $2\pi$, so $\sin(\frac{7\pi}{4})=\sin(2\pi - \frac{\pi}{4})$. According to the property $\sin(2\pi-\alpha)=-\sin\alpha$, we have $\sin(\frac{7\pi}{4})=-\sin(\frac{\pi}{4})$.
Since $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$, then $\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}$.

Step2: Rewrite $\frac{23\pi}{4}$

$\frac{23\pi}{4}=6\pi-\frac{\pi}{4}$. Because the period of the sine - function is $2\pi$, $\sin(\frac{23\pi}{4})=\sin(6\pi-\frac{\pi}{4})$. Using the property $\sin(2k\pi-\alpha)=-\sin\alpha$ ($k = 3$ in this case), we get $\sin(\frac{23\pi}{4})=-\sin(\frac{\pi}{4})$. Since $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$, then $\sin(\frac{23\pi}{4})=-\frac{\sqrt{2}}{2}$.

Answer:

A. $\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}$
A. $\sin\frac{23\pi}{4}=-\frac{\sqrt{2}}{2}$